Let X be an open neighborhood of 0 in Cnwith coordinates (z

A short survey on the Cauchy problem in sheaf theory

Andrea D’Agnolo

Abstract

This short expository paper shows how sheaf theory applies to the study of initial value problems. More precisely, starting with the classical Cauchy-Kowalevski theorem as the only tool borrowed from P.D.E., the methods of the microlocal theory of sheaves of [7] are utilized to obtain “more difficult” results such as the well poseness of the hyperbolic Cauchy problem for hyperfunctions, or of the non- characteristic Cauchy problem with holomorphic ramified data.

AMS subject classification: 35A10, 58G17.

1 Foreword

In the following I will present a short survey on the Cauchy problem in the language of sheaf theory. This should be considered as an appendix to Schneiders’s lectures [11], sketching the interplay between D-modules and the microlocal theory of sheaves by Kashiwara and Schapira [7].

To my knowledge, there does not exist any reference book on this subject.

A good survey (from which I benefitted) may be found in Chapter 11 of [7].

2 Tools from P.D.E.

Let me begin by recalling two basic tools in the study of analytic partial differential equations: the Cauchy-Kowalevski theorem and the Zerner prop- agation lemma.

Let X be an open neighborhood of 0 in Cⁿwith coordinates (z) = (z1, z⁰), and let Y be the hypersurface of X defined by the equation z1 = 0. Let

Appeared in: Bull. Soc. Roy. Sci. Li`ege 63 (1994), no. 3-4, 297–307, Algebraic Analysis Meeting (Li`ege, 1993).

P (z, ∂z) =Pm

|α|=0aα(z)∂_z^α be a differential operator on X with holomorphic coefficients, and denote by σ(P )(z; ζ) =P

|α|=maα(z)ζ^α its principal symbol.

Recall that Y is called non-characteristic for the operator P if,

for every z ∈ Y, σ(P )(z, dz1) 6= 0, (2.1) which is equivalent to the requirement a(m,0...,0)(0, z⁰) 6= 0.

If u is a holomorphic function defined in a neighborhood of Y , one denotes by γY(u) its first m traces on Y , γY(u) = (u|Y, . . . , ∂_z^m−1₁ u|Y).

Theorem 2.1. [Cauchy-Kowalevski] Let v ∈ O_X, (w) ∈ O^m_Y, and consider the Cauchy problem 

P u = v,

γY(u) = (w). (2.2)

Assume that Y is non-characteristic for P . Then (2.2) has a unique solution u ∈ O_X|_Y.

Leray [8] proved a sharper version of the previous result. Denote by B(0, ρ) and B⁰(0, ρ) the open balls of center 0 and radius ρ > 0 in Cⁿ and Cⁿ⁻¹ respectively.

Theorem 2.2. [Cauchy-Kowalevski-Leray] Assume that Y is non-charac- teristic for P . Then there exist constants r, ρ0, δ > 0 such that for every ρ with 0 < ρ ≤ ρ0, every x with |x| ≤ r, and every data v ∈ OX(B(0, ρ)), (w) ∈ O_Y^m(B⁰(0, ρ)), the Cauchy problem (2.2) has a unique solution u ∈ OX(B(0, δρ)).

A useful corollary of the last theorem is the following propagation result:

Lemma 2.3. [Zerner] Let ψ be a C¹ function and set Ω = {x; ψ(x) <

0}. Assume that σ(P )(x0; dψ(x0)) 6= 0 and let u ∈ OX(Ω) be such that P u extends holomorphically to a neighborhood of x0 ∈ ∂Ω. Then u extends holomorphically to a neighborhood of x0.

3 Cauchy-Kowalevski-Kashiwara theorem

Let f : Y −→ X be a morphism of complex manifolds, and consider the associated correspondence of cotangent bundles:

T^∗Y ←−_t

f⁰ Y ×

XT^∗X −→

fπ

T^∗X.

Let M be a coherent DX-module, and recall that f is called non-characteristic for M if

fπ−1(char(M)) ∩^tf⁰⁻¹(T_Y^∗Y ) ⊂ Y ×

X T_X^∗X, (3.1) where T_X^∗X denotes the zero-section of T^∗X.

The following formulation of the Cauchy-Kowalevski theorem is due to Kashiwara [4].

Theorem 3.1. [Cauchy-Kowalevski-Kashiwara] Assume that f is non char- acteristic for M. Then the natural morphism

f⁻¹RHom_DX(M, O_X) −→ RHom_DY(M_Y, O_Y) (3.2) is an isomorphism.

As recalled in [11], Kashiwara’s original proof is based on Theorem 2.1, plus some algebraic arguments. In a sense, the difficult part is the switch of language from the statement of Theorem 2.1, to that of Theorem 3.1.

Let f : Y −→ X be the immersion of the hypersurface defined in the previous section, and let M be the DX-module associated to the operator P :

0 −→ DX

−→ D·P X −→ M −→ 0. (3.3)

In this case,

char(M) = {(z; ζ) ∈ T^∗X; σ(P )(z; ζ) = 0},

and conditions (2.1) and (3.1) are thus equivalent. Recall moreover that Theorems 2.1 and 3.1 are also equivalent. In fact, the free D_X resolution (3.3) gives:

RHom_DX(M, OX) ' [OX

−→ OP · X], and one easily checks that

RHom_DY(M_Y, O_Y) ' (O_Y)^m.

Taking the zero-th and first cohomology group of the isomorphism (3.2), one

then has 

ker P ' (OY)^m, coker P = 0.

In other words, for every v ∈ O_X|_Y, w ∈ (O_Y)^m, one can solve the problems:

 P u = 0

γY(u) = (w) , P u = v, (3.4) and it is evident that (2.2) and (3.4) are equivalent.

4 A statement for Cauchy problems

Let M be a left coherent DX-module, let f : Y −→ X be a morphism of complex manifolds, and assume that f is non-characteristic for M. Let S and T be two left DX and DY-modules respectively, and let

φ : f⁻¹S −→ T (4.1)

be a given DY-linear morphism.

The discussion in the previous section was meant to show the reader that the isomorphism (3.2) is equivalent to the solution of the associated Cauchy problem with holomorphic data. On the same lines, we may say that we will have solved the Cauchy problem with data in S, T , if we establish that the natural morphism:

f⁻¹RHom_DX(M, S) −→ RHom_DY(MY, T ), (4.2) induced by φ, is an isomorphism.

Let K and L be sheaves on X and Y respectively (or, more precisely, ob- jects of the derived categories). As we will see in sections 6.2, 6.3, interesting classes of solution spaces are obtained by considering the situation:

(i) S = RHom(K, OX), (ii) T = RHom(L, O_Y),

(iii) the morphism φ is induced by a morphism φ : f⁻¹K ←− L.

Notice that one has the isomorphisms:

RHom_DX(M, S) ' RHom(K, RHom_DX(M, OX)), RHom_DY(MY, T ) ' RHom(L, RHom_DY(MY, OY)), and recall that Theorem 3.1 gives:

RHom_DY(MY, OY) ' f⁻¹RHom_DX(M, OX).

Setting:

F = RHom_DX(M, OX),

the Cauchy problem (4.2) is thus reduced to giving conditions on f , F , K, L, and φ so that the natural morphism:

f⁻¹RHom(K, F ) −→ RHom(L, f⁻¹F ) (4.3) induced by φ, is an isomorphism.

5 Propagation and micro-support

A statement like (4.3) is almost purely sheaf-theoretical: what is lacking is a sheaf-theoretical counterpart to the notion of non-characteristicity. It is here that the notion of micro-support by Kashiwara and Schapira [7] appears.

With the notations of section 2, Lemma 2.3 says that given g ∈ (OX)x0

and f ∈ (ΓΩOX)x0 such that P f = g|Ω, there exists ef ∈ (OX)x0such that f = f |eΩ and P ef = g. This is equivalent to the vanishing of the first cohomology group of the simple complex associated to the double complex:

(ΓΩOX)x⁰

−→ (ΓP ΩOX)x⁰

↑ ↑

(OX)x0

−→P (OX)x0. Hence, Zerner’s lemma is implied by the requirement:

(RΓ_X\ΩF )_x0 = 0, where F = RHom_DX(DX/DXP, OX).

One is now in a position to forget the complex structure, and to forget P.D.E.

Let X be a real analytic manifold (actually, for most results C¹ manifolds would suffice), and denote by D^b(X) the derived category of the category of bounded complexes of sheaves of C-vector spaces on X.

Definition 5.1. The micro-support SS(F ) of an object F of D^b(X) is the closed conic involutive subset of T^∗X given by: p = (x0; ξ0) /∈ SS(F ) if there exists an open neighborhood U of p such that for every x1 ∈ X and for every C¹ function ψ verifying ψ(x₁) = 0 and dψ(x₁) ∈ U , one has:

(RΓ_ψ≥0F )_x1 = 0.

It is not possible to review the microlocal theory of sheaves here. I shall simply list some elementary results that will be useful in the following.

Let A be a locally closed subset of X, and let M ⊂ X be a closed submanifold. One denotes by CM(A) the Whitney normal cone of A along M . This is a closed conic subset of TMX, the normal bundle to M in X.

One denotes by CA the sheaf which is zero on X \ A, and which is the constant sheaf with stalk C on A. One denotes by D⁰F = RHom(F, CX) the dual of F .

Let f : Y −→ X be a morphism of real analytic manifolds. Similarly to the case of D-modules, one says that f is non-characteristic for F if

fπ−1SS(F ) ∩^tf⁰⁻¹(T_Y^∗Y ) ⊂ Y ×

XT_X^∗X.

One denotes by ωY /X = f^!C_X the relative dualizing complex. Recall that ω_{Y /X} ' or_{Y /X}[dim Y − dim X], where or_{Y /X} denotes the relative orientation sheaf.

Proposition 5.2. (i) Let F ∈ Ob(D^b(X)) and assume that f is non- characteristic for F . Then the following estimate holds:

SS(f⁻¹F ) ⊂^tf⁰fπ−1(SS(F )), and one has a natural isomorphism:

f⁻¹F ⊗ ωY /X

−→ f∼ ^!F.

(ii) Let G ∈ Ob(D^b(Y )) and assume that f is proper on supp(G). Then:

SS(Rf∗G) ⊂ fπtf⁰⁻¹(SS(G)).

(iii) Let F, G ∈ Ob(D^b(X)) and assume that G has R-constructible coho- mology groups, and that SS(F ) ∩ SS(G) ⊂ T_X^∗X. Then:

D⁰G ⊗ F −→ RHom(G, F ).^∼

(iv) Let F ∈ Ob(D^b(X)) and let M ⊂ X be a closed submanifold. Then:

SS(RΓMF ) ⊂ T^∗M ∩ CTM^∗X(SS(F )) (cf. section 6.2 for the identification T^∗M ⊂ T_TM^∗_XT^∗X).

6 Back to P.D.E.

Let X be a complex analytic manifold, let M be a coherent DX-module, and set F = RHom_DX(M, OX). Using Theorem 2.2 it is easy to prove the inclusion:

SS(F ) ⊂ char(M), (6.1)

and in fact equality holds.

As we will now see, using (6.1) and the results of the previous section, it is possible to recover classical results in P.D.E.

Let M be a real analytic manifold and let X be a complexification of M . Recall that the sheaf of Sato’s hyperfunctions is defined by:

BM = RHom(D_X⁰ C_M, OX).

Notice that the natural morphism D_X⁰ F ⊗ G −→ RHom(F, G) applied to F = D_X⁰ C_M, G = O_X gives the embedding

A_M ,→ B_M,

where A_M = C_M ⊗ O_X is the sheaf of real analytic functions on M .

Remark 6.1. More generally, if M is a generic submanifold of a complex manifold X, then RHom(D⁰_XC_M, O_X) is isomorphic to the complex of CR- hyperfunctions (i.e. the complex of hyperfunction solutions on M to the tangential ∂ system).

6.1 Elliptic operators

The short exact sequence:

0 −→ T_M^∗ X −→ M ×

XT^∗X −→ T^∗M −→ 0, defining the conormal bundle T_M^∗ X, and the isomorphism:

M ×XT^∗X ' T^∗M ⊕ iT^∗M,

which is due to the complex structure, give an identification:

T_M^∗X ' iT^∗M.

Recall that an operator P ∈ D_X is called elliptic if σ(P )(x; iξ) 6= 0 for every ξ 6= 0.

More generally, a coherent D_X-module M is called elliptic if char(M) ∩ T_M^∗X ⊂ T_X^∗X.

Applying Proposition 5.2 (iii) to F = RHom_DX(M, OX) and G = D⁰C_M, one thus gets the classical regularity theorem:

Theorem 6.2. Let M be an elliptic coherent DX-module. Then:

RHom_DX(M, AM)−→ RHom^∼ _DX(M, BM).

6.2 Hyperbolic operators

One of the keys to understand hyperbolic operators is the identification of T^∗M to a submanifold of T_T^∗∗

MXT^∗X.

Let −H : T^∗T^∗X −→ T T^{∼ ∗}X be the opposite of the Hamiltonian iso- morphism H. If Λ ⊂ T^∗X is a Lagrangian submanifold, −H induces an identification T^∗Λ ' TΛT^∗X. In particular, for Λ = T_M^∗ X one gets:

T^∗T_M^∗ X ' TTM^∗XT^∗X ' T_T^∗∗

MXT^∗X. (6.2)

By the zero section M −→ T_M^∗ X of π : T_M^∗X −→ M , and by the induced map:

tπ⁰ : T_M^∗X ×

MT^∗M −→ T^∗T_M^∗X, one gets a map:

T^∗M ' M ×

MT^∗M −→ T^∗T_M^∗ X. (6.3) Composing (6.2) and (6.3) one gets the identification:

T^∗M ,→ T_T^∗∗

MXT^∗X. (6.4)

Let N be a real submanifold of M , let Y ⊂ X be a complexification of N , and consider the embeddings:

Y −→^f X

↑^j ↑ⁱ

N −→ M.^g

One says that N is hyperbolic for a coherent DX-module M if

T_N^∗M ∩ CTM^∗X(char(M)) ⊂ T_M^∗M (6.5) via the identification (6.4).

Of course, if M = D_X/D_XP is the D_X-module associated to a single differential operator P ∈ DX, one recovers the usual definition of (weak) hyperbolicity.

As proved by [1], [7], the good solution space for the study of hyperbolic Cauchy problems is the space of Sato’s hyperfunctions.

Theorem 6.3. Assume that N is hyperbolic for M. Then the natural mor- phism:

f⁻¹RHom_DX(M, B_M) −→ RHom_DY(M_Y, B_N) (6.6) is an isomorphism.

Notice that (6.6) is of the form (4.2). It also fit with (4.3) if one sets F = RHom_DX(M, O_X), K = D⁰C_M, L = D⁰C_N, and let φ be the morphism induced by the natural morphism CM −→ CN.

Proof. By Proposition 5.2 (iv), hypothesis (6.5) implies that g is non-char- acteristic for RΓM(F ). One then has the chain of isomorphisms:

g⁻¹RΓM(F ) ' g^!RΓM(F ) ⊗ ω_N/M^⊗−1 ' g^!i^!F ⊗ ω_N/M^⊗−1 ' j^!f^!F ⊗ ω_N/M^⊗−1 ' RΓN(F ) ⊗ ω_N/M^⊗−1.

Recall that D⁰C_M ∼= ωM/X. We have thus proved the isomorphism:

f⁻¹RHom_DX(M, BM) ' RHom_DX(M, ΓNBM) ⊗ ω_N/M^⊗−1. The proof is then achieved by following “division” lemma.

Lemma 6.4. Assume that Y is non-characteristic for M. Then there is an isomorphism:

RHom_DX(M, ΓNBM) ' RHom_DY(MY, BN) ⊗ ωN/M. (6.7) Proof. Setting F = RHom_DX(M, OX), one has the isomorphisms:

RΓN(F ) ' j^!f^!F

' RΓ_N(f⁻¹F ) ⊗ ω_{Y /X}.

By Theorem 3.1, f⁻¹F ' RHom_DY(MY, OY), and one concludes.

6.3 Ramified functions

An instance of naturally occurring solution spaces when studying non-cha- racteristic Cauchy problems with singular data (or characteristic problems with holomorphic data) is the space of ramified holomorphic functions.

Let p : fC^∗ −→ C be the universal covering of C^∗ = C \ {0}. Recall that one can choose a coordinate t ∈ C ' fC^∗ so that p(t) = exp(2πit).

Let Y be a germ of smooth hypersurface of a complex manifold X, and choose a locally defined complex analytic function g : X −→ C, with dg 6= 0, such that Y = g⁻¹(0). Set fX^∗ = fC^∗ ×C X, and consider the Cartesian diagram:

Xf^∗ −→ fC^∗

↓^pX ↓^p X −→ C.^g

The complex O_{Y /X}^ram of ramified holomorphic function along Y in X is defined as O_{Y /X}^ram = Rp_{X ∗}p⁻¹_X OX.

Notice that p^!_X is an exact functor and that p^!_X = p⁻¹_X . By the Poincar´e- Verdier duality one gets:

Rp_{X ∗}p⁻¹_X OX = RHom(Rp_{X !}C_f

X^∗, OX)

= RHom(g⁻¹p!C_f

C∗, OX).

Summarizing up, one can write

O_{Y /X}^ram = RHom(K, O_X), (6.8) where K = g⁻¹p!C_f

C∗.

Remark 6.5. More generally, let T ⊂ X be a closed subset, ep : eX −→ X \ T be a covering map, and set p = i ◦ ep, where i : X \ T −→ X is the inclusion map. The complex RHom(K, OX), for K = p!C_e

X, may then be considered as a complex of “ramified” holomorphic functions along T .

Let X be an open subset of Cⁿ with 0 ∈ X, let z = (z1, z⁰) = (z1, . . . , zn) be the coordinates on X and let (z; ζ) be the associated coordinates in T^∗X.

Consider the Cauchy problem:

 P u = 0,

γY(u) = (w), (6.9)

where P is a differential operator of order m, the hyperplane Y = {z ∈ X; z1 = 0} is non-characteristic for P , and (w) are holomorphic functions on Y ramified along the hypersurface Z = {z ∈ X; z1 = z2 = 0}. Let f : Y −→ X be the embedding. Suppose that P has characteristics with constant multiplicities transversal to Y ×X T^∗X at^tf⁰⁻¹(T_Z^∗Y ) ∩ char(M).

Let Z1, . . . , Zrbe the (locally uniquely determined) smooth hypersurfaces of X whose conormal bundles are the union of the bicharacteristics of P issued from^tf⁰⁻¹( ˙T_Z^∗Y ). Notice that the Zi are transversal to each other, and transversal to Y .

In [3], Hamada, Leray and Wagschal proved that the holomorphic solution of (6.9), defined in a neighborhood of Y \ Z, extends holomorphically as a sum of ramified functions along the Zj’s.

Following [2], let us reduce this statement to the form (4.2), (4.3).

Choose analytic functions g : Y −→ C, gi : X −→ C with dg 6= 0, dgi 6= 0, such that gi◦ f = g and Z = g⁻¹(0), Zi = g⁻¹_i (0). Set L = g⁻¹p!C_e

X, K_i = g_i⁻¹p_!C_e

X, and let K be the third term of a distinguished triangle:

K −→ ⊕^r_i=1Ki

−→ ⊕h ^r−1_i=1C_X −→,⁺¹

where h is the composite of the map ⊕^r_i=1τj and the map ⊕^r_i=1C_X −→

⊕^r−1_i=1C_X, given by (a₁, . . . , a_r) 7→ (a₂ − a₁, . . . , a_r− a_r−1). Notice that the natural morphisms L −→ CY, Ki −→ CX, induce a morphism

φ : L −→ f⁻¹K.

The sheaf of holomorphic functions on Y ramified along Z is given by:

O_Z/Y^ram = RHom(L, OY), and one may check that the complex

X

i

O_Z^ram_i/X = RHom(K, OX)

is concentrated in degree zero, and that its sections are sums of ramified functions along the Zj’s.

Theorem 6.6. ([2]) The natural morphism:

f⁻¹RHom_DX(M,X

i

O_Z^rami/X)|Z −→ RHom_DY(MY, O_Z/Y^ram)|Z (6.10)

is an isomorphism.

Proof. With the above notations, we may rewrite (6.10) as f⁻¹RHom(K, F )|Z −→ RHom(L, f∼ ⁻¹F )|Z,

and it is possible to give a purely sheaf-theoretical proof of this isomorphism for which we refer to [2].

References

[1] J. M. Bony, P. Schapira, Solutions hyperfonctions du probl`eme de Cauchy. Lecture Notes in Math., Springer 287 (1973)

[2] A. D’Agnolo, P. Schapira, An inverse image theorem for sheaves with applications to the Cauchy problem. Duke Mathematical Journal, 64 No. 3 (1991)

[3] T. Hamada, J. Leray, C. Wagschal, Syst`emes d’´equations aux d´eriv´ees partielles `a caract´eristiques multiples: probl`eme de Cauchy ramifi´e, hy- perbolicit´e partielle. J. Math. Pures et Appl. 55 (1976)

[4] M. Kashiwara, Algebraic study of systems of partial differential equa- tions. Thesis (in japanese), Tokyo (1971)

[5] M. Kashiwara, P. Schapira, Probl`eme de Cauchy pour les syst`emes mi- crodifferentiels dans le domain complexe. Inventiones Math. 46 (1978) [6] M. Kashiwara, P. Schapira, Microhyperbolic systems. Acta Math. 142

(1979)

[7] M. Kashiwara, P. Schapira, Sheaves on manifolds. Springer Berlin Hei- delberg New York 292 (1990)

[8] J. Leray, Probl`eme de Cauchy I. Bull. Soc. Math. France 85 (1957) [9] M. Sato, T. Kawai, M. Kashiwara, Hyperfunctions and pseudo-differen-

tial equations. In Hyperfunctions and pseudo-differential equations. H.

Komatsu ed., Proceedings Katata 1971. Lect. Notes in Math. Springer Berlin Heidelberg New York 287 (1973)

[10] P. Schapira, Microdifferential systems in the complex domain. Springer Berlin Heidelberg New York 269 (1985)

[11] J.-P. Schneiders, Introduction to D-modules. In these proceedings.